1 placement optimization of energy and information access

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1

Placement Optimization of Energy and Information

Access Points in Wireless Powered Communication

Networks

Suzhi Bi, Member, IEEEand Rui Zhang,Senior Member, IEEE

Abstract

The applications of wireless power transfer technology to wireless communications can help build a wireless

powered communication network (WPCN) with more reliable and sustainable power supply compared to the

conventional battery-powered network. However, due to thefundamental differences in wireless information and

power transmissions, many important aspects of conventional battery-powered wireless communication networks

need to be redesigned for efficient operations of WPCNs. In this paper, we study the placement optimization of

energy and information access points in WPCNs, where the wireless devices (WDs) harvest the radio frequency

energy transferred by dedicated energy nodes (ENs) in the downlink, and use the harvested energy to transmit

data to information access points (APs) in the uplink. In particular, we are interested in minimizing the network

deployment cost with minimum number of ENs and APs by optimizing their locations, while satisfying the energy

harvesting and communication performance requirements ofthe WDs. Specifically, we first study the minimum-

cost placement problem when the ENs and APs are separately located, where an alternating optimization method

is proposed to jointly optimize the locations of ENs and APs.Then, we study the placement optimization when

each pair of EN and AP are co-located and integrated as a hybrid access point, and propose an efficient algorithm

to solve this problem. Simulation results show that the proposed methods can effectively reduce the network

deployment cost and yet guarantee the given performance requirements, which is a key consideration in the future

applications of WPCNs.

Index Terms

Wireless power transfer, wireless powered communication networks, energy harvesting, network planning,

node placement optimization.

This work will be presented in part at the IEEE Global Communications Conference (GLOBECOM), San Diego, CA, USA, Dec. 6-10,2015. This work was supported in part by the National NaturalScience Foundation of China (Project number 61501303).

S. Bi is with the College of Information Engineering, Shenzhen University, Shenzhen, Guangdong, China 518060 (e-mail:[email protected]).

R. Zhang is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, and alsowith the Institute for Infocomm Research, A∗STAR, Singapore 138632 (e-mail:[email protected]).

December 10, 2015 DRAFT

http://arxiv.org/abs/1505.06530v2

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I. INTRODUCTION

Modern wireless communication systems, e.g., cellular networks and wireless sensor networks (WSNs),

are featured by larger bandwidth, higher data rate and lowercommunication delays. The improvement on

communication quality and the increased data processing complexity have imposed higher requirement on

the quality of power supply to wireless devices (WDs). Conventionally, WDs are powered by batteries,

which have to be replaced/recharged manually once the energy is depleted. Alternatively, the recent

advance of radio frequency (RF) enabled wireless power transfer (WPT) provides an attractive solution

to power WDs over the air [1], [2]. By leveraging the far-fieldradiative properties of microwave, WDs

can harvest energy remotely from the RF signals radiated by the dedicated energy nodes (ENs) [3].

Compared to the conventional battery-powered methods, WPTcan save the cost due to manual battery

replacement/recharging in many applications, and also improve the network performance by reducing

energy outages of WDs. Currently, tens of microwatts (µW) RF power can be effectively transferred to

a distance of more than10 meters.1 The energy is sufficient to power the activities of many low-power

communication devices, such as sensors and RF identification (RFID) tags. In the future, we expect more

practical applications of RF-enabled WPT to wireless communications thanks to the rapid developments

of many performance enhancing technologies, such as energybeamforming with multiple antennas [4]

and more efficient energy harvesting circuit designs [5].

In a wireless powered communication network (WPCN), the operations of WDs, including data trans-

missions, are fully/partially powered by means of RF-enabled WPT [6]–[14]. A TDMA (time division

multiple access) based protocol for WPCN is first proposed in[6], where the WDs harvest RF energy

broadcasted from a hybrid access point (HAP) in the first timeslot, and then use the harvested energy

to transmit data back to the HAP in the second time slot. Later, [7] extends the single-antenna HAP in

[6] to a multi-antenna HAP that enables more efficient energytransmission via energy beamforming as

well as more spectrally efficient SDMA (space division multiple access) based information transmission

as compared to TDMA. To further improve the spectral efficiency, [8] considers using full-duplex HAP

in WPCNs, where a HAP can transmit energy and receive user data simultaneously via advanced self-

interference cancelation techniques. Intuitively, usinga HAP (or co-located EN and information AP),

instead of two separated EN and information access point (AP), to provide information and energy access

is an economic way to save deployment cost, and the energy andinformation transmissions in the network

1Based on the product specifications on the website of Powercast Co. (http://www.powercastco.com), with TX91501-3W power transmitterand P2110 Powerharvester receiver, the harvest RF power at adistance of10 meters is about40 µW.


http://www.powercastco.com

3

can also be more efficiently coordinated by the HAP. However,using HAP has an inherent drawback that

it may lead to a severe “doubly-near-far” problem due to distance-dependent power loss [6]. That is, the

far-away users quickly deplete their batteries because they harvest less energy in the downlink (DL) but

consume more energy in the uplink (UL) for information transmission. To tackle this problem, separately

located ENs and APs are considered to more flexibly balance the energy and information transmissions

in WPCNs [9]–[11]. In this paper, we consider the method using either co-located or separate EN and

information AP to build a WPCN.

Most of the existing studies on WPCNs focus on optimizing real-time resource allocation, e.g., transmit

signal power, waveforms and time slot lengths, based on instantaneous channel state information (CSI,

e.g., [6]–[8]). In this paper, we are interested in the long-term network performance optimization based

mainly on the average channel gains. It is worth mentioning that network optimizations in the two

different time scales are complementary to each other in practice. That is, we use long-term performance

optimization methods for the initial stage of network planning and deployment, while using short-term

optimization methods for real-time network operations after the deployment. Many current works on

WPCNs use stochastic models to study the long-term performance because of the analytical tractability,

especially when the WDs are mobile in location. For instance, [9] applies a stochastic geometry model

in a cellular network to derive the expression of transmission outage probability of WDs as a function of

the densities of ENs and information APs. Similar stochastic geometry technique is also applied to WPT-

enabled cognitive radio network in [10] to optimize the transmit power and node density for maximum

primary/secondary network throughput. However, in many application scenarios, the locations of the WDs

are fixed, e.g., a sensor network with sensor (WD) locations predetermined by the sensed objects, or an

IoT (internet-of-things) network with static WDs. In this case, a practical problem that directly relates

to the long-term performance of WPCNs, e.g., sensor’s operating lifetime, is to determine the optimal

locations of the ENs and APs. Nonetheless, this important node placement problem in WPCNs is still

lacking of concrete studies.

In conventional battery-powered wireless communication networks, node placement problem concerns

the optimal locations of information APs only, which has been well investigated especially for wireless

sensor networks using various geometric, graphical and optimization algorithms (see e.g., [15]–[19]).

However, there exist major differences between the node placement problems in battery-powered and

WPT-enabled wireless communication networks. On one hand,a common design objective in battery-

powered wireless networks is to minimize the highest transmit power consumption among the WDs to


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satisfy their individual transmission requirements. However, such energy-conservation oriented design is

not necessarily optimal for WPCNs, because high power consumption of any WD can now be replenished

by means of WPT via deploying an EN close to the WD. On the otherhand, unlike information

transmission, WPT will not induce harmful co-channel interference to unintended receivers, but instead

can boost their energy harvesting performance. These evident differences indicate that the node placement

problem in battery-powered wireless communication networks should be revisited for WPCNs, to fully

capture the advantages of WPT.

In this paper, we study the node placement optimization problem in WPCNs, which aims to minimize

the deployment cost on ENs and APs given that the energy harvesting and communication performances

of all the WDs are satisfied. Our contributions are detailed below.

1) We formulate the optimal node placement problem in WPCNs using either separated or co-located

EN and AP. To simplify the analysis, we then transform the minimum-cost deployment problem

into its equivalent form that optimizes the locations of fixed number of ENs and APs;

2) The node placement optimization using separated EN and APis highly non-convex and hard to

solve. To tackle the non-convexity of the problem, we first propose an efficient cluster-based greedy

algorithm to optimize the locations of ENs given fixed AP locations. Then, a trial-and-error based

algorithm is proposed to optimize the locations of APs givenfixed ENs locations. Based on the

obtained results, we further propose an effective alternating method that jointly optimizes the EN

and AP placements;

3) For the node placement optimization using co-located EN and AP (or HAP), we extend the greedy

EN placement method under fixed APs to solving the HAP placement optimization, which is

achieved by incorporating additional considerations of dynamic WD-HAP associations during HAP

placement. Specifically, a trial-and-error method is used to solve the WD-HAP association problem,

which eventually leads to an efficient greedy HAP placement algorithm.

Due to the non-convexity of the node placement problems in WPCNs, all the proposed algorithms are

driven by the consideration of their applicabilities to large-size WPCNs, e.g., consisting of hundreds of

WDs and EN/AP nodes. Specifically, we show that the proposed algorithms for either separated or co-

located EN and AP placement are convergent and of low computational complexity. Besides, simulations

validate our analysis and show that the proposed methods caneffectively reduce the network deployment

cost to guarantee the given performance requirements. The proposed algorithms may find their wide

application in the future deployment of WPCNs, such as wireless sensor networks and IoT networks.


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Energy

receiver

Information

transmitter

Power supply

EN1

AP1

AP2

AP3

WD1 WD2

WDK

WD3

Energy flow Information flow

EN2

f1

f2

Fig. 1. Schematics of a WPCN with separate ENs and APs.

The rest of the paper is organized as follows. In Section II, we first introduce the system models

of WPCN where the ENs and APs are either separated or co-located. Then, we formulate the optimal

node placement problems for the two cases in Section III, andpropose efficient algorithms to solve the

problems in Sections IV and V, respectively. In Section VI, simulations are performed to evaluate the

performance of the proposed node placement methods. Finally, we conclude the paper and discuss future

working directions in Section VII.

II. SYSTEM MODEL

A. Separated ENs and APs

For the case of separated ENs and APs, we consider in Fig. 1 a WPCN in R2 consisting ofM

ENs,N APs andK WDs, whose locations are denoted by2× 1 coordinate vectors{ui|i = 1, · · · ,M},

{vj|j = 1, · · · , N}, and {wk|k = 1, · · · , K}, respectively. We assume that the energy and information

transmissions are performed on orthogonal frequency bandswithout interfering with each other. Specifi-

cally, the ENs are connected to stable power source and broadcast RF energy in the DL for the WDs to

harvest the energy and store in their rechargeable batteries. At the same time, the WDs use the battery

power to transmit information to the APs in the UL. The circuit structure of a WD to perform the above

operations is also shown in Fig. 1.

In a transmission block of lengthT , theM ENs transmit simultaneously on the same bandwidth in


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the DL, where each ENi transmits

xi(t) =√

P0si(t), t ∈ [0, T ] , i = 1, · · · ,M. (1)

Here,P0 denotes the transmit power,si(t) denotes the pseudo-random energy signal used by thei-th EN,

which is assumed to be of unit power (Et [|si(t)|2] = 1) and independent among the ENs (Et [si(t)sj(t)] =

0 if i 6= j). The reason to use random signal instead of a single sinusoid tone is to avoid peak in transmit

power spectrum density, for satisfying the equivalent isotropically radiated power (EIRP) requirement

enforced by spectrum regulating authorities [1]. Notice that the energy beamforming technique proposed

in [4] is not used in our setup, as it requires accurate CSI andDL symbol-level synchronization, which

may be costly to implement in a highly distributed WPCN network considered in this work.

Accordingly, the received energy signal by thek-th WD is

yk(t) =√

P0

∑Mi=1αi,ksi(t), k = 1, · · · , K, (2)

whereαi,k denotes the equivalent baseband channel coefficient from the i-th EN to thek-th WD, which is

assumed to be constant within a transmission block but may vary independently across different blocks.

Let hi,k , |αi,k|2 denote the channel power gain, which follows a general distribution with its mean

determined by the distance between the EN and WD, i.e.,

E [hi,k] = β||ui −wk||−dD , i = 1, · · · ,M, k = 1, · · ·K, (3)

where dD ≥ 2 denotes the path loss exponent in DL,|| · || denotes thel2-norm operator, andβ ,

Ad

(

3·108

4πfd

)dDwith Ad andfd denoting the downlink antenna power gain and carrier frequency, respectively

[20]. Then, each WDk can harvest an average amount of energy from the energy transmission within

each block given by [3]

Qk = ηTE[

|yk(t)|2| {hi,k}]

= ηTP0

(

∑Mi=1hi,k

)

, k = 1, · · · , K, (4)

where η ∈ (0, 1] denotes the energy harvesting circuit efficiency, and the expectation is taken over

the pseudo-random energy signal variations under fixedhi,k’s within the transmission block. Letλk ,

E [Qk] /T denote the average energy harvesting rate over the variation of wireless channels (hi,k’s) in

different transmission blocks, we have

λk = ηβP0 ·∑M

i=1||ui −wk||−dD , k = 1, · · · , K. (5)

In the UL information transmissions, we assume that each WD transmits data to only one of the APs.


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To make the placement problem tractable, the WD-AP associations are assumed to be fixed, where each

WD k transmits to its nearest APjk regardless of the instantaneous CSI, i.e.,

jk = arg minj=1,··· ,N

||vj −wk||, k = 1 · · · , K. (6)

Here, we assume no co-channel interference for the receiveduser signals from different WDs, e.g., the

WDs transmit on orthogonal channels. Besides, for the simplicity of analysis, we assume no limit on the

maximum number of WDs that an AP could receive data from. Then, the average power consumption

rate for WDk is modeled as

µk = a1,k + Ek , a1,k + a2,k||vjk −wk||dU , k = 1, · · · , K, (7)

where a1,k denotes the constant circuit power of WDk, Ek denotes the average transmit power as a

function of the distance between WDk and its associated APjk, anddU ≥ 2 denotes the UL channel

path loss exponent. Besides,a2,k > 0 denotes a parameter related to the transmission strategy used in

the UL communication.2 In general, the model in (7) indicates that the transmit power increases as a

polynomial function of the distance between the transmitter and receiver to satisfy certain communication

quality requirement, e.g., minimum data rate or maximum allowed outage probability, which is widely

used for wireless network performance analysis [21], [22].

B. Co-located ENs and APs

A special case of the WPCN that we consider in Fig. 1 is when theENs and APs are grouped into

pairs and each pair of EN and AP are co-located and integratedas a hybrid access point (HAP), which

corresponds to settingM = N and ui = vi for i = 1, · · · ,M . With the network model and HAP’s

circuit structure shown in Fig. 2, a HAP transfers RF power inthe DL and receives information in the

UL simultaneously on different frequency bands. Although the use of HAPs is less flexible in placing

the ENs and APs than with separated ENs and APs, the overall deployment cost is reduced, because the

production and operation cost of a HAP is in general less thanthe sum-cost of two separate EN and AP.

For brevity, we reuse the notationui, i = 1, · · · ,M , to denote the location coordinates of theM HAPs.

Given other parameters unchanged, the expression of the average energy harvesting rateλk of the k-th

2For example, given a receive signal power requirementΓk to achieve a target data rate and maximum allowed outage portability ψk forWD k, we havea2,k = Γk

Au

(

4πfu3·108

)dUE1

(

ln(

11−ψk

))

when truncated channel inverse transmission [20] is used under Rayleigh fading

channel, whereAu and fu denote the uplink antenna gain and carrier frequency, respectively, andE1(x) ,∫

∞

11t· e−tx dt denotes the

exponential integral function.


8

HAP1HAP2

HAP3

WD1

WD2

WDK

WD3

Energy

transmitter

Information

receiver

f1

f2

Energy flow Information flow

Fig. 2. Schematics of a WPCN with co-located ENs and APs (HAPs).

WD is the same as that in (5). Meanwhile, the average power consumption rateµk can be obtained from

(7) by replacingvj with ui as follows.

µk = a1,k + a2,k||uik −wk||dU , k = 1, · · · , K, (8)

whereik is the index of the HAP that WDk associates with, i.e.,

ik = arg mini=1,··· ,M

||ui −wk||, k = 1, · · · , K. (9)

C. Net Energy Harvesting Rate

With the above definitions, thenet energy harvesting rates of the WDs in both cases of separate and

co-located EN and AP are given by

ωk = λk − µk, k = 1, · · · , K. (10)

In practice, the net energy harvesting rate can directly translate to the performance of device operating

lifetime (see e.g., [23]). Specifically, given an initial battery levelC, the average time before thek-th

WD’s battery depletes is−C/ωk whenωk < 0, and+∞ whenωk ≥ 0.3 In other words, given a minimum

3We neglect in this paper the battery degradation effect [24]caused by repeated charging and discharging.


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device operating lifetime requirementTk > 0, it must satisfyωk ≥ −C/Tk if Tk < ∞, andωk ≥ 0 if

Tk = ∞.

III. PROBLEM FORMULATION

In this paper, we assume that the locations of the WDs are known and study the optimal placement

of ENs and information APs, which are either separated or co-located in their locations. This may

correspond to a sensor network with sensor (WD) locations predetermined by the sensed objects, or an

IoT network with static WDs. In particular, we are interested in minimizing the deployment cost given

that the net energy harvesting rates of all the WDs are largerthan a common prescribed valueγ, i.e.,

ωk ≥ γ, k = 1 · · · , K, whereγ is set to achieve a desired device operating lifetime.

A. Separated ENs and APs

When the ENs and APs are separated, the total deployment costis c1M + c2N if M ENs andN APs

are used, wherec1 and c2 are the monetary costs of deploying an EN and an AP, respectively. To solve

the minimum-cost deployment problem, let us first consider the following feasibility problem:

Find UM = [u1, · · · ,uM ] , V

N = [v1, · · · ,vN ] (11a)

s. t. λk

(

UM)

− µk

(

VN)

≥ γ, k = 1, · · · , K, (11b)

bl ≤ ui ≤ b

h, i = 1, · · · ,M, (11c)

bl ≤ vj ≤ b

h, j = 1, · · · , N, (11d)

whereλk andµk are functions ofUM andVN given in (5) and (7), respectively. The inequalities in (11c)

and (11d) denote element-wise relations. Besides,{

bl,bh

}

specifies a feasible deployment area for both

the ENs and APs inR2, which is large enough to contain all the WDs, i.e.,bl ≤ wk ≤ b

h, k = 1 · · · , K.

Evidently, if (11) can be efficiently solved for anyM andN , then the optimal node placement solution

to the considered minimum-cost deployment problem can be easily obtained through a simple two-

dimension search over the values ofM andN , i.e., finding a pair of feasible(M,N) that produces the

lowest deployment costc1M + c2N .

For a pair of fixedM andN (M > 0 andN > 0), we can see that (11) is feasible if and only if the


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optimal objective of the following problem is no smaller than γ, i.e.,

maxUM ,VN

mink=1,··· ,K

{

λk

(

UM)

− µk

(

VN)}

(12a)

s. t. bl ≤ ui ≤ b

h, i = 1, · · · ,M, (12b)

bl ≤ vj ≤ b

h, j = 1, · · · , N. (12c)

We can express (12) as its equivalent epigraphic form [27], i.e.,

maxt,UM ,VN

t

s. t. λk

(

UM)

− µk

(

VN)

≥ t, k = 1, · · · , K,

bl ≤ ui ≤ b

h, i = 1, · · · ,M,

bl ≤ vj ≤ b

h, j = 1, · · · , N.

(13)

Given fixedM andN , (11) is feasible if and only if the optimal objective of (13)satisfiest∗ ≥ γ. Then,

the key difficulty of solving the optimal deployment problemis to find efficient solution for problem

(13).

B. Co-located ENs and APs

When the ENs and APs are integrated as HAPs, the total deployment cost isc3M if M HAPs are

used. Here,c3 denotes the cost of deploying a HAP, where in generalc3 < c1 + c2. Similar to the case

of separated ENs and APs, the minimum-cost placement problem can be equivalently formulated as the

following feasibility problem for any fixed number ofM > 0 HAPs,

Find UM = [u1, · · · ,uM ]

s. t. λk

(

UM)

− µk

(

UM)

≥ γ, k = 1, · · · , K,

bl ≤ ui ≤ b

h, i = 1, · · · ,M,

(14)

whereλk

(

UM)

andµk

(

UM)

are given in (5) and (8), respectively. Notice that the studyon co-located

ENs and APs is not a special case of that of separated ENs and APs. In fact, it adds extra constraints

(ui = vi, i = 1, · · · ,M) to (11), which leave less flexibility to the nodes placementdesign and make

the problem more challenging to solve. Equivalently, the feasibility of (14) can be determined by solving


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the following optimization problem

maxt,UM

t

s. t. λk

(

UM)

− µk

(

UM)

≥ t, k = 1, · · · , K,

bl ≤ ui ≤ b

h, i = 1, · · · ,M,

(15)

and then comparing the optimal objectivet∗ with γ, to see whethert∗ ≥ γ holds. In the following

Sections IV and V, we propose efficient algorithms to solve problems (13) and (15), respectively. It is

worth mentioning that the placement solution to (13) and (15) can be at arbitrary locations. When an

EN (or a HAP) is placed at a location very close to an WD, the far-field channel model in (3) may

be inaccurate. However, we learn from (13) and (15) that the optimal value t∗ is determined by the

performance-bottleneck WD that is far away from the ENs and APs (i.e., the channel model in (3)

applies practically), thus having very low energy harvesting rate and high transmit power consumption.

Therefore, the potential inaccuracy of (3) will not affect the objective values of (13) and (15), and the

proposed algorithms in this paper are valid in practice.

IV. PLACEMENT OPTIMIZATION OF SEPARATED ENS AND APS

In this section, we study the node placement optimization for separately located EN and AP in problem

(13). Specifically, we first study in Section IV.A the method to optimize EN placement assuming that

the locations of APs are fixed in a WPCN. In Section IV.B, we further study the method to optimize the

placement of APs given fixed EN locations. Based on the obtained results, we then propose in Section

IV.C an alternating method to jointly optimize the placements of ENs and APs. In addition, an alternative

local searching method is considered in Section IV.D for performance comparison.

A. EN Placement Optimization with Fixed AP Location

We first consider the optimal EN placement problem when the locations of the APs are fixed, i.e.,

vj ’s are known. In this case, the WD-AP associationjk is known for each WDk from (6), andµk’s can

be calculated accordingly from (7). It is worth mentioning that the proposed algorithms under the fixed

AP setup can be directly extended to solve EN placement problem in other wireless powered networks

not necessarily for communication purpose, e.g., a sensor network whose energy is mainly consumed on

sensing and processing data, as long as the energy consumption ratesµk’s are known parameters. With


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vj ’s andµk’s being fixed, we can rewrite (13) as

maxt,UM

t

s. t. ϕ ·∑Mi=1||ui −wk||−dD − µk ≥ t, k = 1, · · · , K,

bl ≤ ui ≤ b

h, i = 1, · · · ,M,

(16)

whereϕ , ηβP0. We can see that (16) is a non-convex optimization problem, because||ui −wk||−dD is

neither a convex nor concave function inui. As it currently lacks of effective method to convert (16) into

a convex optimization problem, the optimal solution is in general hard to obtain. However, for a special

case withM = 1, i.e., placing only one EN, the optimal solution is obtainedin the following. By setting

M = 1, (16) can be rewritten as

maxt,u1

t (17a)

s. t. ||u1 −wk||dD ≤ ϕ

t+ µk, k = 1, · · · , K, (17b)

bl ≤ u1 ≤ b

h. (17c)

Although (17) is still a non-convex optimization problem (as ϕ/(t+ µk) is not a concave function int),

it is indeed a convex feasibility problem overu1 when t is fixed, which can be efficiently solved using

the interior point method [27]. Therefore, the optimal solution of (17) can be obtained using a bi-section

search method overt, whose pseudo-code is given in Algorithm 1. Notice that the right hand side (RHS)

of (17b) is always positive during the bisection search overt ∈ (LB1, UB1). Besides, we can infer

that Algorithm 1 converges to the optimal solutiont∗, because problem (17) is feasible fort ≤ t∗ and

infeasible otherwise. The total number of feasibility tests performed islog2 [(UB1 − LB1) /σ1], where

σ1 is a predetermined parameter corresponding to a solution precision requirement.

Since placing one EN optimally is solved, we have the potential to decouple the difficult EN placement

problem (16) intoM relatively easy problems withM > 1. This motivates a greedy algorithm, which

places the ENs iteratively one-by-one into the network. Intuitively, an optimal deployment solution of

(16) should “spread” theM ENs among theK WDs to maximize the minimum energy harvesting rate.

However, the optimal solution obtained from solving (17) tends to place the single EN around the center

of the cluster formed by theK WDs. Inspired by the optimal solution structures of (16) and(17), we

propose in the following a cluster-based greedy EN placement method, where the newly placed EN

optimizes the net energy harvesting rates of an expanding cluster of WDs, until all the WDs are included.


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Algorithm 1: Bi-section search for single EN placement.input : WD locationswk ’s, power consumption ratesµk ’soutput: the optimal location of the ENu∗

1

1 Initialize: LB1 ← −maxk=1,··· ,K µk, UB1 ← P0;2 repeat3 t← (UB1 + LB1)/2;4 if Problem (17) is feasible givent then5 LB1 ← t;6 u1 ← a feasible solution of (17) givent;7 else8 UB1 ← t;9 end

10 until |UB1 − LB1| < σ1;11 Return u

∗ ← the last feasible solutionu1;

In practice, we geographically partition theK WDs intoM non-overlapping clusters (assumingK ≥M), denoted by{Wi|i = 1, · · · ,M}.4 This can be efficiently achieved by, e.g., the well-knownK-means

clustering algorithm [25]. Then, in thei-th iteration (i ≥ 1), we obtain the optimal location of thei-th

EN, denoted byu∗

i , by maximizing the net energy harvesting rates of the WDs in the first i clusters as

follows.

maxti,ui

ti (18a)

s. t. (ti + µk − λi−1,k) · ||ui −wk||dD ≤ ϕ,

k ∈ {W1 ∪ · · · ∪Wi} , (18b)

bl ≤ ui ≤ b

h, (18c)

whereλi−1,k denotes the accumulative RF power harvested at thek-th WD due to the(i− 1) previously

deployed ENs, given by

λi−1,k =

0, i = 1,

ϕ · ∑i−1j=1||u∗

j −wk||−dD , i > 1.(19)

In each iteration,u∗

i can be efficiently obtained using a bi-section search methodover ti similar to that

in solving (17). The pseudo-code of the greedy algorithm is given in Algorithm 2. Notice in line8 of

Algorithm 2, the corresponding inequality in (18b) holds automatically whenti + µk − Qi−1,k < 0 for

somek, given a fixedti ∈ (LB2, UB2). Therefore, the corresponding constraints can be safely ignored

without affecting the feasibility of (18).

4The proposed node placement algorithms can apply to any clustering method used. Besides, the algorithm complexity is not related tothe clustering method as long as the partitions of the WDs aregiven. For simplicity, we useK-means clustering algorithm in this paper.


14

AP3

AP1

AP4

AP2

AP3

AP1

AP2EN1

AP3

AP1

AP4

AP2EN1

EN2

AP3

AP1

AP4

AP2EN1

EN2

EN3

(a) (b)

(c) (d)

AP4

WDs not considered in the current EN deployment

WDs considered in the current EN deployment

the EN currently being deployed

ENs already deployed

Fig. 3. Illustration of greedy algorithm for placingM = 3 ENs (with fixed APs).

The greedy EN placement method is illustrated in Fig. 3. In this example, we first divide the WDs

into M = 3 clusters in Fig. 3(a), then place the3 ENs one-by-one in Fig. 3(b)-(d). When placing the

1st EN or EN1, the algorithm only considers the received energy of the WDsin the 1st cluster (shaded

WDs in Fig. 3(b)) from the EN to be placed; for the2nd EN or EN2, it considers the received energy

of the WDs in the1st and2nd clusters from the first2 ENs; for the last EN or EN3, it considers the

received energy of all the WDs from the3 ENs. Notice that our greedy algorithm allows multiple ENs

to be placed in the same cluster, because the placement of thei-th EN considers all the WDs in the first

i clusters.

Algorithm 2 applies Algorithm 1M times, one for placing each EN, thus the total number of

feasibility tests performed isM log2 [(UB2 − LB2) /σ2], whereσ2 is a parameter corresponding to a

solution precision requirement. Besides, the time complexity of solving each convex feasibility test using

the interior point method isO(√

K + 2M log (K + 2M))

[27]. Therefore, the overall time complexity

of Algorithm 2 is O(

M√K + 2M log (K + 2M)

)

, which is moderate even for a large-size network

consisting of, e.g., tens of ENs and hundreds of WDs.


15

Algorithm 2: Greedy algorithm forM EN placement.input : WD locationswk ’s, N AP locationsvj ’s;output: locations ofM ENs {u∗

1, · · · ,u∗

M};1 Initialization: Clustering the WDs into{Wi, i = 1, · · · ,M} ;2 With vj ’s, calculateµk ’s using (6) and (7);3 for i = 1 to M do4 LB2 ← −maxk=1,··· ,K µk, UB2 ←MP0 ;5 Update

{

Qi−1,k, k = 1 · · · ,K}

using (19);6 repeat7 ti ← (UB2 + LB2)/2;8 Ignore the constraints in (18b) withti + µk − Qi−1,k < 0;9 if Problem (18) is feasible giventi then

10 LB2 ← ti;11 u

∗

i ← a feasible solution of (18);12 else13 UB2 ← ti;14 end15 until |UB2 − LB2| < σ2;16 end17 Return {u∗

1, · · · ,u∗

M}

B. AP Placement Optimization under Fixed ENs

We then study in this subsection the method to optimize the placement of APs given fixed EN locations,

i.e., ui’s are known. In this case,λk’s are fixed and can be calculated using (5). Withλk’s being fixed

parameters, we can substitute (7) into (13) and formulate the optimal AP placement problem under fixed

ENs as followsmaxt,VN

t

s. t. λk − a1,k − a2,k||vjk −wk||dU ≥ t, k = 1, · · · , K,

bl ≤ vj ≤ b

h, j = 1, · · · , N,

(20)

wherejk is the index of AP that WDk associates with given in (6). The above problem is non-convex

because of the combinatorial nature of WD-AP associations,i.e., jk’s are discrete indicators. However,

notice that ifjk’s are known, (20) is a convex problem that is easily solvable. In practice, however,jk’s

are revealed only after (20) is solved and the placement of APs is obtained. To resolve this conflict, we

propose in the following atrial-and-error method to find feasiblejk’s and accordingly a feasible AP

placement solution to (20). The pseudo-code of the method tosolve (20) is presented in Algorithm 3 and

explained as follows.

As its name suggests, we first convert (20) into a convex problem by assuming a set of WD-AP

associations, denoted byj(b)k , k = 1, · · · , K, and then solve (20) for the optimal AP placement based on

the assumedj(b)k ’s. Next, we comparej(b)k ’s with the actual WD-AP associations after the optimal AP


16

placement is obtained using (6), denoted byj(a)k , k = 1, · · · , K. Specifically, we check ifj(a)k = j

(b)k , ∀k.

If yes, we have obtained a feasible solution to (20); otherwise, we updatej(b)k = j(a)k , k = 1, · · · , K

and repeat the above process untilj(a)k = j

(b)k , ∀k. The convergence of Algorithm 3 is proved in the

Appendix and the convergence rate is evaluated numericallyin Fig. 6 of Section VI. Intuitively, the

trial-and-error method is convergent because the optimal value of (20) is bounded, while by updating

j(b)k = j

(a)k , we can always improve the optimal objective value of (20) inthe next round of solving it.

As we will show later in Fig. 6 of Section VI, the number of iterations used until convergence is of

constant order, i.e.,O(1), regardless of the value ofN or K. There, the time complexity of Algorithm

3 is O(√

K + 2N log (K + 2N))

, as it takes this time complexity for solving (20) in each iteration.

Algorithm 3: Trial-and-error method forN AP placementinput : K WD locationswk ’s, M EN locationsui’s;output: locations ofN APs, i.e.,{v∗

1 , · · · ,v∗

N};1 Initialization:2 Separate the WDs intoN clusters, and place each AP at a cluster center. Usev

(0)j ’s to denote the initial AP locations;

3 With v(0)j ’s, calculatej(b)k ’s using (6);

4 With ui’s, calculateλk ’s using (5). Letflag ← 1;5 while flag = 1 do6 Given j(b)k ’s, solve (20) for optimal AP placementv∗

j ’s;7 Given v

∗

j ’s, calculatej(a)k ’s using (6);8 if j(a)k 6= j

(b)k for somek then

9 Updatej(b)k = j(a)k , k = 1, · · · ,K;

10 else11 A local optimum is found, return{v∗

1 , · · · ,v∗

N};12 flag ← 0;13 end14 end

C. Joint EN and AP Placement Optimization

In this subsection, we further study the problem of joint EN and AP placement optimization. In this

case, we consider both the locations of ENs and APs as variables, such that the joint EN-AP placement


17

problem in (13) can be expressed as

maxt,UM ,VN

t

s. t. ϕ ·∑Mi=1||ui −wk||−dD − a2,k||vjk −wk||dU

≥ t + a1,k, k = 1, · · · , K,

bl ≤ ui ≤ b

h, i = 1, · · · ,M

bl ≤ vj ≤ b

h, j = 1, · · · , N.

(21)

Evidently, the optimization problem is highly non-convex because of the non-convex function||ui −wk||−dD and the discrete variablesjk’s. Based on the results in Section IV.A and IV.B, we propose an alter-

nating method in Algorithm 4 to solve (21) for joint EN and AP placement solution. Specifically, starting

with a feasible AP placement, we alternately apply Algorithms 2 and 3 to iteratively update the locations

of ENs and APs, respectively. A point to notice is that Algorithm 2 (and Algorithm 3) only produces a

sub-optimal solution to (16) (and (20)), thus the objectivevalue of (21) may decrease during the alternating

iterations. To cope with this problem, we record the deployment solutions obtained inL > 1 iterations and

select the one with the best performance. The impact of the parameterL to the algorithm performance is

evaluated in Fig. 5 of Section VI. Given the complexities of Algorithms 2 and 3, we can easily infer that

the time complexity of Algorithm 4 isO(

LM√K + 2M log (K + 2M) + L

√K + 2N log (K + 2N)

)

.

Algorithm 4: An alternating method for joint AP-EN placement.input : K WD locationswk ’s, L iterations;output: Locations ofM ENs andN APs, i.e.,ui’s andvj ’s.;

1 Initialize: Separate the WDs intoN clusters, and place each AP at a cluster center. Usevj ’s to denote the initial AP locations;2 for l = 1 to L do3 if l is odd then4 Given vj ’s, solve (21) forui’s using Algorithm 2;5 else6 Givenui’s, solve (21) forvj ’s using Algorithm 3;7 end8 zl ← mink=1,··· ,K (λk − µk), whereλk andµk are in (5) and (7), respectively.;9 u

(l)i ← ui, i = 1, · · · ,M , andv(l)

j ← vj , j = 1, · · · , N .10 end11 p← argmaxl=1,··· ,L zl;12 Return: u(p)

i ’s andv(p)j ’s.

D. Alternative Method

Besides the proposed alternating method for solving (21), we also consider an alternative local searching

method used as benchmark algorithm for performance comparison. The local searching algorithm starts


18

with a random deployment of theM ENs andN APs, i.e.,ui’s andvi’s, and checks if the minimum net

energy harvesting rate among the WDs, i.e.,

Pr , mink=1,··· ,K

(

ϕ ·∑Mi=1||ui −wk||−dD − a1,k − a2,k min

j=1,··· ,N||vj −wk||dU

)

(22)

can be increased by making a random movement toui’s and vj ’s that satisfy{

ui, vj, i = 1, · · · ,M, j = 1, · · · , N,

∣

∣

∣

∣

∑Mi=1||ui − ui||2 +

∑Nj=1||vj − vj||2 < σ3

}

, (23)

whereσ3 is a fixed positive parameter. If yes, it makes the move and repeats the random movement

process. Otherwise, ifPr cannot be increased, the algorithm has reached a local maximum and returns the

current placement solution. Several off-the-shelf local searching algorithms are available, where simulated

annealing [26] is used in this paper. In particular, simulated annealing can improve the searching result

by allowing the nodes to be moved to locations with decreasedvalue ofPr to reduce the chance of being

trapped at local maximums. Besides, we can improve the quality of deployment solution using different

initial node placements, which are obtained either randomly or empirically, and select the resulted solution

with the best performance.

V. PLACEMENT OPTIMIZATION OF CO-LOCATED ENS AND APS

In this section, we proceed to study the node placement optimization problem (15) for the case of

co-located ENs and APs. The problem is still non-convex due to which the optimal solution is hard to be

obtained. Inspired by both Algorithms 2 and 3, we propose in this section an efficient greedy algorithm

for HAP placement optimization.

A. Greedy Algorithm Design

The node placement optimization problem (15) is highly non-convex, because the expression of problem

(15) involves non-convex function||ui−wk||−dD in λk

(

UM)

and minimum operator over convex functions

in µk

(

UM)

. Since its optimal solution is hard to obtain, a promising alternative is the greedy algorithm,

which iteratively places a single HAP to the network at one time, similar to Algorithm 2 for solving

(16) which optimizes the EN locations given fixed APs. However, by comparing problems (15) and (16),

we can see that the algorithm design for solving (15) is more complicated, because eachµk is now a

function ofui’s, instead of constant parameter in (16).

Similar to the greedy algorithm in Section IV.A, we first separate theK WDs intoM non-overlapping

clusters, denoted by{Wi|i = 1, · · · ,M}, and add to the network a HAP in each iteration. Specifically,


19

in the i-th iteration, given that the previous(i − 1) HAPs are fixed, we obtain the optimal location of

the i-th HAP, denoted byu∗

i , by maximizing the net energy harvesting rates of the WDs in the first i

clusters. To simplify the notations, we also useλi−1,k as in Section IV.A to denote the accumulative RF

harvesting power of the WDk from the previously placed(i− 1) HAPs, which can be calculated using

(19). Besides, letµi−1,k denote the energy consumption rate of thek-th WD after the first(i− 1) HAPs

have been placed, where

µi−1,k =

+∞, i = 1,

a1,k + a2,k minj=1,··· ,i−1 ||u∗

j −wk||dU , i > 1.(24)

Notice that the only difference between placing thei-th HAP and thei-th EN in Section IV.A is thatµk

is now a function ofui instead of a given constant. By substituting (24) into (18),the optimal location

of the i-th HAP is obtained by solving the following problem

maxti,ui

ti (25a)

s. t. (ti + µi,k − λi−1,k) · ||ui −wk||dD ≤ ϕ,

k ∈ {W1 ∪ · · · ∪ Wi} , (25b)

ul ≤ ui ≤ u

h, (25c)

where

µi,k = min(

µi−1,k, a1,k + a2,k||ui −wk||dU)

. (26)

From (26), we can see that a WD may change its association to the i-th HAP, if the newly placed HAP is

closer to the WD than all the other(i−1) HAPs that have been previously deployed. This combinatorial

nature of WD-AP associations makes problem (25) non-convexeven if ti is fixed. In the following, we

apply the similar trial-and-error technique as that in Section IV.B to obtain a feasible solution to problem

(25).

B. Solution to Problem (25)

The basic idea to obtain a feasible solution of (25) is to convert it into a convex problem giventi,

and then use simple bi-section search overti. The convexification of (25) is achieved by a trial-and-error

method similar to that used for finding feasible WD-AP associations proposed in Algorithm 3. That is, we

iteratively make assumptions on WD-AP associations and update the optimal placement of thei-th HAP

obtained from solving (25) based on the assumptions in the current iteration. With a bit abuse of notations,


20

here we reuseu∗

i in each iteration as the optimal location of thei-th HAP given the current WD-AP

association assumptions. Specifically, we assume whether the WDs change their associations after thei-th

HAP is added, i.e., assuming eitherµi−1,k < a1,k + a2,k||u∗

i −wk||dU or µi−1,k ≥ a1,k + a2,k||u∗

i −wk||dU

for eachk. Then, given a fixedti, each constraint onk in (25b) belongs to one of the following four

cases:

1) Case1: If we assume that WDk does not change its WD-HAP association after thei-th HAP is

placed into the WPCN, or equivalentlyµi−1,k < a1,k+a2,k||u∗

i −wk||dU , we can replace the corresponding

constraint in (25b) with

(ti + µi−1,k − λi−1,k) · ||ui −wk||dD ≤ ϕ. (27)

With a fixed ti, (27) is a convex constraint ifti + µi−1,k − λi−1,k > 0.

2) Case2: If we still assumeµi−1,k < a1,k + a2,k||u∗

i −wk||dU , while ti + µi−1,k − λi−1,k ≤ 0 holds,

we can safely drop the constraint in (25b) without changing the feasible region ofui.

3) Case3: On the other occasion, if we assume that WDk changes its WD-HAP association, or

µi−1,k ≥ a1,k + a2,k||u∗

i −wk||dU , the corresponding constraint in (25b) becomes

(

ti + a1,k + a2,k||ui −wk||dU − λi−1,k

)

· ||ui −wk||dD ≤ ϕ, (28)

which can be further expressed as

||ui −wk||dU+dD +ti + a1,k − λi−1,k

a2,k||ui −wk||dD − ϕ

a2,k≤ 0. (29)

Notice that, given a fixedti, (29) is a convex constraint ifti + a1,k − λi−1,k ≥ 0.

4) Case4: Otherwise, if we assumeµi−1,k ≥ a1,k + a2,k||u∗

i − wk||dU and ti + a1,k − λi−1,k < 0

holds, (29) is a non-convex constraint, as the left-hand-side (LHS) of (29) is the difference of two convex

functions. Nonetheless, we show that (29) can still be converted into a convex constraint in this case. Let

us first consider a function

z(x) = xdU+dD +ti + a1,k − λi−1,k

a2,kxdD − ϕ

a2,k, (30)

wherex ≥ 0 and ti + a1,k − λi−1,k < 0. We calculate the first order derivative ofz(x) and find thatz

increases monotonically when

x >

[− (ti + a1,k − λi−1,k) dDa2,k (dU + dD)

]1/dU

, τi,k, (31)

and decreases monotonically ifx ≤ τi,k. Notice thatτi,k > 0 andz(0) = − ϕa2,k

< 0 always hold. Therefore,

althoughz(x) is not a convex function,z(x) < 0 can still be equivalently expressed asx < θi,k, with θi,k


21

being some positive number satisfyingz(θi,k) = 0. The value ofθi,k can be efficiently obtained using

many off-the-shelf numerical methods, such as the classic Newton’s method or bi-section search method.

A close comparison between the LHS of (29) andz(x) in (30) shows that, by lettingx , ||ui − wk||,we can equivalently express (29) as a convex constraint

||ui −wk|| ≤ θi,k, (32)

when ti + a1,k − λi−1,k < 0 holds.

To sum up, given a fixedti, we tackle thek-th constraint in (25b) using one of the following methods:

1) Replace by (27) if assumingµi−1,k < a1,k + a2,k||u∗

i −wk||dU and ti + µi−1,k − λi−1,k > 0;

2) Drop the constraint if assumingµi−1,k < a1,k + a2,k||u∗

i −wk||dU and ti + µi−1,k − λi−1,k ≤ 0;

3) Replace by (29) if assumingµi−1,k ≥ a1,k + a2,k||u∗

i −wk||dU and ti + a1,k − λi−1,k ≥ 0;

4) Replace by (32) if assumingµi−1,k ≥ a1,k + a2,k||u∗

i −wk||dU and ti + a1,k − λi−1,k < 0.

After processing all theK constraints in (25b), we can convert (25) into a convex feasibility problem given

a set of WD-HAP association assumptions and a fixedti. Accordingly, the optimal placement of thei-th

HAP (u∗

i ) under the assumptions, can be efficiently obtained from solving (25) using a bi-section search

method overti. Similar to the trial-and-error technique used in Algorithm 3, we check if the obtained

u∗

i satisfies all the assumptions made. If yes, we have obtained afeasible solution of (25). Otherwise,

we switch the violating assumptions, then follow the above constraint processing method to resolve (25)

for a newu∗

i , and repeat the iterations until all the assumptions are satisfied. The above trial-and-error

method converges. The proof follows the similar argument asgiven in the Appendix, which proves the

convergence of the trial-and-error method used for solvingproblem (20). Thus, this proof is omitted here.

C. Overall Algorithm

Since the placement of a single HAP can be obtained via solving (25), we can iteratively place theM

HAPs into the WPCN. The pseudo-code of the revised greedy algorithm is presented in Algorithm 5. For

example, Fig. 4 illustrates the detailed steps taken to place the2nd HAP, or HAP2 (total 3 HAPs while

the first HAP, or HAP1 is already placed). Specifically, we first assume in Fig. 4(a)that all the WDs in

the 1st cluster associate with HAP1, and the WDs in the2nd cluster associate with HAP2, after HAP2 is

added into the network. Then, we obtain in Fig. 4(b) the optimal placement of the HAP2 based on the

association assumptions made. However, the obtained location of HAP2 results in a contradiction with

the association assumption made on WD1 (assumed to be associated with HAP1). Therefore, we change


22

HAP1

(a) (b)

(d)

Assumed WD-to-HAP association

Optimal HAP location under association assumption

HAP1

(c)

HAP2

Contradicting

assumption

HAP1

Change

association

assumption

Association

assumptions

New

assumptions

HAP1

All

assumptions

valid

Placement

solution for

HAP2

HAP2

HAP2

HAP2

Actual WD-to-HAP association

WD1 WD1

WD1WD1

HAP location before optimization

Fig. 4. Illustration of greedy algorithm for placement optimization with co-located EN and AP (HAP).

the association assumption of WD1 to HAP2, and recalculate the optimal placement solution for HAP2

(Fig. 4(c)). In Fig. 4(d), the newly obtained location of HAP2 satisfies all the association assumptions,

thus the placement of HAP2 is feasible. Following the similar argument in the Appendix, the association

assumption update procedure converges, because the optimal objective value of (25) is non-decreasing

upon each association assumptions update (lines10−27 of Algorithm 5). After obtaining the location of

HAP2, a feasible location of the3rd HAP, HAP3, can also be obtained using the similar procedures as

above. Besides, we can infer that the time complexity of Algorithm 5 isO(

M√K + 2M log (K + 2M)

)

,

because it places theM HAPs iteratively, while the trial-and-error method used toplace each HAP needs

O(√

K + 2M log (K + 2M))

complexity.

VI. SIMULATION RESULTS

In this section, we use simulations to evaluate the performance of the proposed node placement methods.

All the computations are executed by MATLAB on a computer with an Intel Core i52.90-GHz CPU and

4 GB of memory. The carrier frequency is915 MHz for both DL and UL transmissions operating on

different bandwidths. In the DL energy transmission, we consider using Powercast TX91501-1W power


23

Algorithm 5: Greedy algorithm for HAP placement optimization.input : K WD locationswk ’soutput: locations ofM HAPs {u∗

1, · · · ,u∗

M}1 Cluster the WDs into{Wi, i = 1, · · · ,M};2 for i = 1 to M do3 for each WDk do4 Updateλi−1,k andµi−1,k using (19) and (24);5 Assume WDk satisfies condition(a) or (b):6 (a) µi−1,k < a1,k + a2,k||u

∗

i −wk||dU ;

7 (b) µi−1,k ≥ a1,k + a2,k||u∗

i −wk||dU .

8 end9 StopF lag ← 0;

10 repeat11 LB ← −δ, UB ← δ, δ is sufficiently large;12 repeat13 ti ← (UB + LB)/2;14 Given ti, convert (25) into a convex problem using the procedures in Section V.B;15 if Problem (25) is feasible giventi then16 LB ← ti; u∗

i ← a feasible solution of (25);17 else18 UB ← ti;19 end20 until |UB − LB| < σ, σ is sufficiently small;21 if all theK assumptions are validthen22 StopF lag ← 1; the i-th HAP location← u

∗

i ;23 else24 StopF lag ← 0;25 For each WD violating the assumption, switch the assumptionfrom (a) to (b), or (b) to (a);26 end27 until StopF lag = 1;28 end29 Return the HAP locations{u∗

1, · · · ,u∗

M}.

transmitter withP0 = 1W (Watt) transmit power, and P2110 receiver withAd = 3 dB antenna gain

and η = 0.51 energy harvesting efficiency. Besides, we assume the path loss exponentdD = 2.2, thus

β = 6.57 × 10−4. In the UL information transmission, we assume thatdU = 2.5, a1,k = 50µW and

a2,k = 1.4 · 10−6 for k = 1, · · · , K, wherea2,k is obtained assuming Rayleigh fading and the use of

truncated channel inversion transmission [20] with receiver signal power−70dBm (equivalent to18dB

SNR target with -88dBm noise power) and outage probability of5%. All the WDs, ENs and APs are

placed within a24m× 24m box region specified bybl = (0, 0)T andbh = (24, 24)T . Unless otherwise

stated, each point in the following figures is an average performance of20 random WD placements, each

with K = 60 WDs uniformly placed within the box region.

A. Separated EN and AP Deployment

We first evaluate the performance of the proposed alternating optimization method (Algorithm 4) for

placing separated ENs and APs. Without loss of generality, we considerN = 6 APs and show in Fig.


24

4 6 8 10 12 14 16 18 20 22 24−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

No. of ENs (M)

Min

imum

net

ene

rgy

harv

estin

g ra

te (

Pr)

/mW

AltOpt (L=20)AltOpt (L=10)LSOnly EN OptimizedCC

Fig. 5. Performance comparison of the separated AP and EN placement methods (N = 6).

5 the minimum net energy harvesting ratePr in (15) achieved by Algorithm 4 when the locations of

APs are jointly optimized with those of different number of ENs (M). Evidently, a largerPr indicates

better system performance. For the proposed alternating optimization algorithm (AltOpt), we show both

the performance withL = 10 and 20. Besides, we also consider the following benchmark placement

methods

• Cluster center method (CC): separate the WDs intoM clusters and place an EN at each of the cluster

centers. Similarly, separate the WDs intoN clusters and place theN APs at the cluster centers;

• Optimize only EN locations: the APs are placed at theN cluster centers; while the EN placement

is optimized based on the AP locations using Algorithm 2.

• Local searching algorithm (LS) method introduced in Section IV, where the initial EN and AP

locations are set according to the CC method and the best-performing deployment solution obtained

during the searching iterations is used.

Evidently, we can see that the proposed alternating optimization has the best performance among the

methods considered. Specifically, significant performancegain is observed for AltOpt over optimizing

EN placement only. The LS method has relatively good performance compared to AltOpt, especially

whenM is small, but the performance gap increases withM due to the increasing probability of being


25

4 6 8 10 12 14 16 18 20 22 240

1

2

3

4(a) Average no. of iterations till convergence (K=60 WDs)

No. of APs (N)N

o. o

f ite

ratio

ns

40 50 60 70 80 900

1

2

3

4(b) Average no. of iterations till convergence (N=8 APs)

No. of WDs (K)

No.

of i

tera

tions

Fig. 6. The average number of WD-AP association assumptionsmade before convergence of Algorithm 3, as a function of (a)N underfixedK = 60; and (b) as a function ofK under fixedN = 8.

trapped at local maximums with a largerM . The CC scheme has the worst performance as it neglects the

disparity of energy harvesting/consumption rates among the WDs and precludes the case where multiple

AP/ENs can be placed in a cluster. In practice, Fig. 5 can be used to evaluate the deployment cost of each

algorithm. For instance, whenPr ≥ −0.1mW is required, we see that the AltOpt (L = 10) on average

needs6 ENs, the LS method needs8 ENs, optimizing EN placement only requires9 ENs, and the CC

method needs20 ENs, with the same number of information APs deployed (i.e.,N = 6). The above

results show that, when the ENs and APs are separated, significant performance gain can be obtained by

jointly optimizing the placements of ENs and APs, especially for large-size WPCNs that need a large

number of ENs and APs to be deployed. In addition, as optimizing only EN locations corresponds to

a special case of Algorithm 4 withL = 1, we can see that the performance gain is significant whenL

increases from1 to 10. However, the performance improvement becomes marginal aswe further increase

L from 10 to 20. In practice, good system performance can be obtained with relatively small number of

alternating optimizations, e.g.,L = 10 in our case.

We then show in Fig. 6 the convergence rate of Algorithm 3, forwhich the convergence is proved in

an asymptotic sense in the Appendix. In particular, we plot the average number of iterations (WD-AP

association assumptions) used until the algorithm converges. Here, we investigate the convergence rate


26

4 6 8 10 12 14 16 18 20 22 24

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

No. of HAPs (M)

Min

imum

net

ene

rgy

harv

estin

g ra

te (

Pr)

/mW

GreedyLSCC

Fig. 7. Performance comparison of the co-located AP and EN (HAP) placement methods.

when either the number of APs (N) or WDs (K) varies. With fixedK in Fig. 6(a), we see that the number

of iterations used till convergence does not vary significantly asN increases. Similarly in Fig. 6(b), with

a fixedN = 8, we do not observe significant increase of iterations whenK increases from40 to 90.

Besides, all the simulations performed in Fig. 6 use at most7 iterations to converge. Therefore, we can

safely estimate that the number of iterations used till convergence is of constant order, i.e.,O(1), which

leads to the complexity analysis of Algorithm 4 in Section IV.C and Algorithm 5 in Section V.C.

B. Co-located EN and AP Deployment

Next, we evaluate in Fig. 7 the performance of the proposed Algorithm 5 for co-located ENs and APs,

where the value ofPr achieved by Algorithm 5 is plotted against the number of HAPsused (M). In

particular, we compare its performance with that of LS (withM cluster centers as the initial searching

points) and the CC placement method, i.e., the HAPs are placed at theM cluster centers. We can see that

the proposed greedy algorithm in Algorithm 5 has the best performance among the methods considered.

Nonetheless, the performance gaps over the LS and CC methodsare relatively small compared to that

in Fig. 5. An intuitive explanation is that the doubly-near-far phenomenon for co-located EN and AP

renders the optimal HAPs placement to be around the cluster centers. By comparing Fig. 5 and Fig. 7,

we can see the evident performance advantage of using separated ENs and APs over HAPs. For instance,


27

4 6 8 10 12 14 16 18 20 22 240

5

10

15

20

25

30

35

40

No. of HAPs (M)

Nor

mal

ized

CP

U ti

me

GreedyLS

Fig. 8. Comparison of CPU time of the LS and Greedy methods forHAP placement optimization.

the Pr achieved by6 ENs and6 APs in Fig. 5 is−0.1mW, while that achieved by6 HAPs (equivalent

to 6 ENs and6 APs being co-located) is only around−0.17mW.

Although the greedy algorithm and the LS method perform closely, they differ significantly in the

computational complexity. To better visualize the growth rate of complexity, we plot in Fig. 8 the

normalized CPU time of the two methods, where each point on the figure is the normalized against

the CPU time of the respective method whenM = 4. Clearly, we can see that the complexity of the

greedy algorithm increases almost linearly withM , where the CPU time increases approximately6 times

whenM increases from4 to 24. This matches our complexity analysis for Algorithm 5 in Section V.C

that the complexity increases almost linearly inM whenK is much larger. The LS method, however,

has a much faster increase in complexity withM , where the CPU time increases by around39 times

whenM increases from4 to 24. Therefore, even in a large-size WPCN with largeM , the computation

time of the proposed greedy algorithm is still moderate, while this may be extremely high for the LS

method, e.g., couple of minutes versus several hours forM = 50.

C. Case Study: Separated Versus Co-located EN and AP

Finally, we present a case study to compare the cost of node placement achieved by using either

separated or co-located ENs and APs. Here, we consider a WPCNwith 60 WDs uniformly placed within


28

−0.15 −0.1 −0.05 0 0.050

10

20

30

40

(a) Minimum net energy harvesting requirement (γ) / mWD

eplo

ymen

t cos

t/uni

t

−0.15 −0.1 −0.05 0 0.050

10

20

30

40

(b) Minimum net energy harvesting requirement (γ) / mW

Dep

loym

ent c

ost/u

nit

SeparatedCo−located (c

3=1.4)

SeparatedCo−located (c

3=1.05)

7 HAPs10 HAPs

13 HAPs

13 HAPs

19 HAPs

19 HAPs

24 HAPs

(M=4,N=5)

(M=4,N=5)

(M=9,N=4)

(M=9,N=4)

(M=14,N=4)(M=19,N=5)

(M=26,N=4)

(M=19,N=5)(M=14,N=4)

(M=26,N=4)

24 HAPs

10 HAPs7 HAPs

Fig. 9. Comparison of the minimum deployment costs achievedby separated/colocated EN and AP placement optimization methods, wherein (a) c3 = 1.4 and in (b)c3 = 1.05.

the 24m× 24m box region, where the detailed placement is omitted due to the page limit. For the case

of separated ENs and APs, we use Algorithm 4 to enumerate(M,N) pairs that can satisfy a given net

energy harvesting performance requirementγ, and select the one with the minimum cost as the solution.

For the case of HAPs, we use Algorithm 5 to find the minimumM that can satisfy the performance

requirementγ. A point to notice is that, the obtained deployment solutions are sub-optimal to the min-

cost deployment problems with either separated or co-located ENs and APs, i.e., the cost of the optimal

deployment solution can be lower, because Algorithms 4 and 5are sub-optimal to solve problems (13)

and (15), respectively. More effort to further improve the solution performance is needed for future

investigations.

In Fig. 9(a), we show the minimum deployment costs achieved by the two methods under different

performance requirementγ. The number of nodes used by both methods are also marked in the figure.

With {c1, c2, c3} = {0.7, 1, 1.4}, we can see that using separated ENs and APs can achieve much lower

deployment cost than co-located HAPs. In another occasion in Fig. 9(b), the two scheme achieve similar

deployment cost when the cost of a HAP is decreased fromc3 = 1.4 to 1.05. We also observe that, by

allowing the ENs and APs to be separately located, we need much less energy/information access points

than they are co-located to achieve the same performance, thanks to the extra freedom in choosing both


29

the numbers and locations of energy/information access points. For instance, whenγ = 0, we need24

separated energy/information access points (M = 19 andN = 5), while 38 co-located energy/information

access points by the19 HAPs. However, we do not intend to claim that using separatedENs and APs is

better than the co-located case. Rather, we show that a cost-effective deployment plan can be efficiently

obtained using the proposed methods. In practice, the choice of using either separated or co-located

ENs and APs depends on a joint consideration of the node deployment costs, the network size and the

performance requirement.

VII. CONCLUSIONS AND FUTURE WORK

In this paper, we studied the node placement optimization problem in WPCN, which aims to minimize

the node deployment cost while satisfying the energy harvesting and communication performance of

WDs. Efficient algorithms were proposed to optimize the nodeplacement of WPCNs with the ENs and

information APs being either separated and co-located. In particular, when ENs and APs are separately

located, simulation results showed that significant deployment cost can be saved by jointly optimizing

the locations of ENs and APs, especially for large-size WPCNs with a large number of ENs and APs

to be deployed. In the case of co-located ENs and APs (HAPs), however, the performance advantage of

node placement optimization is not that significant, where we observed relatively small performance gap

between optimized node placement solutions and that achieved by some simple heuristics, i.e., placing

the HAPs at the cluster centers formed by the WDs. In practice, separated ENs and APs are more suitable

for deploying WPCNs than co-located HAPs, because of the flexibility in choosing both the numbers and

locations of ENs and APs. Nonetheless, because the optimal solution to the node placement problem has

not been obtained in this paper, we may expect further improvement upon our proposed methods in the

future, especially for the case of HAP placement optimization.

Finally, we conclude with some interesting future work directions for the node placement problem in

WPCN. First, the models considered in this paper can be extended to more general setups. Using the uplink

information transmission as example, we assumed in this paper that each WD has fixed association with

a single AP. In practice, dynamic frequency allocation can be applied to enhance the spectral efficiency,

where a WD can transmit to different APs, even multiple APs simultaneously, in different transmission

blocks. Besides, instead of assuming each AP can serve infinite number of WDs, we can allow each

AP to serve finite number of WDs. In addition, we may also consider the presence of uplink co-channel

interference due to frequency reuse in WPCN. The extensionsrequire adding corresponding constraints


30

or changing the expression of energy consumption model in the problem formulation of this paper.

Second, it is interesting to consider the hybrid node deployment problem that uses both co-located and

separated ENs/APs. Third, it is practically important to consider the node placement problem with location

constraints, e.g., some areas that may forbid the ENs/APs tobe placed. In addition, the density of ENs

may be constrained to satisfy certain safety considerationon RF power radiation.

APPENDIX

PROOF OF THECONVERGENCE OFALGORITHM 3

Let{

t(l),v(l)j , j = 1, · · · , N

}

denote the optimal solutions of (20) calculated from thel-th (l ≥ 1) set

of assumptions made on the WD-AP associations, denoted byj(l)k , k = 1 · · · , K. Let K(l) denote the set

of WDs to which the optimal solutionv(l)j ’s contradict with the WD-AP assumptions (we consider only

K(l) 6= ∅, because otherwise the algorithm has reached its optimum),i.e.,{

K(l): k = 1 · · · ,K, ||v

j(l)k

−wk|| > minj=1,··· ,N

||v(l)j −wk||

}

. (33)

According to the proposed trial-and-error method,j(l+1)k is set for eachk = 1, · · · , K as

j(l+1)k =

j(l)k , k /∈ K(l),

argminj=1,··· ,N ||v(l)j −wk||, k ∈ K(l).

(34)

Let t(l+1) denote the minimum net energy harvesting rate among all the WDs given the updated WD-AP

associationj(l+1)k ’s and the current AP locationsv(l)

j ’s, i.e.,

t(l+1) = mink=1·,K

(

λk − a1,k − a2,k||vj(l+1)k

−wk||dU)

. (35)

We can see thatt(l+1) ≥ t(l) because the update of WD-AP associations in (34) does not increase the energy

consumption rate of any WD achived by assumingj(l)k , k = 1 · · · , K. Besides,

{

t(l+1),v(l)j , j = 1, · · · , N

}

is a feasible solution of (20) under the association assumption j(l+1)k ’s. Therefore, the optimal solution

{

t(l+1),v(l+1)j , j = 1, · · · , N

}

calculated from the association assumptionj(l+1)k ’s will lead to t(l+1) ≥

t(l+1) ≥ t(l). In other words, the optimal objective of (20) is non-decreasing in each trial-and-error update

of WD-AP associations. This, together with the fact that theoptimal value of (20) is bounded, leads to

the conclusion that the proposed trial-and-error method isconvergent.

REFERENCES

[1] S. Bi, C. K. Ho, and R. Zhang, “Wireless powered communication: opportunities and challenges,”IEEE Commun. Mag., vol. 53, no. 4,

pp. 117-125, Apr. 2015.


31

[2] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RF energy harvesting: a contemporary survey,”IEEE

Commun. Surveys Tuts., vol. 17, no. 2, pp. 757-789, 2015.

[3] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information andpower transfer: architecture design and rate-energy tradeoff,” IEEE Trans.

Commun., vol. 61, no. 11, pp. 4754-4767, Nov. 2013.

[4] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,”IEEE Trans. Wireless Commun.,

vol. 12, no. 5, pp. 1989-2001, May 2013.

[5] A. Georgiadis, G. Andia, and A. Collado, “Rectenna design and optimization using reciprocity theory and harmonic balance analysis

for electromagnetic (EM) energy harvesting,”IEEE Antennas Wireless Propag. Lett., vol. 9, pp. 444-446, May 2010.

[6] H. Ju and R. Zhang, “Throughput maximization in wirelesspowered communication networks,”IEEE Trans. Wireless Commun., vol. 13,

no. 1, Jan. 2014.

[7] L. Liu, R. Zhang, and K. C. Chua, “Multi-antenna wirelesspowered communication with energy beamforming,”IEEE Trans. Commun.,

vol. 62, no. 12, pp. 4349-4361, Dec. 2014.

[8] H. Ju and R. Zhang, “Optimal resource allocation in full-duplex wireless powered communication network,”IEEE Trans. Commun.,

vol. 62, no. 10, pp. 3528-3540, Oct. 2014.

[9] K. Huang and V. K. N. Lau, “Enabling wireless power transfer in cellular networks: architecture, modeling and deployment,” IEEE

Trans. Wireless Commun., vol. 13, no. 2, pp. 902-912, Feb. 2014.

[10] S. Lee, R. Zhang, and K. B. Huang, “Opportunistic wireless energy harvesting in cognitive radio networks,”IEEE Trans. Wireless

Commun., vol. 12, no. 9, pp. 4788-4799, Sept. 2013.

[11] Y. Che, L. Duan, and R. Zhang, “Spatial throughput maximization of wireless powered communication networks,”IEEE J. Sel. Areas

Commun., vol. 33, no. 8, pp. 1534-1548, Aug. 2015.

[12] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Wireless-powered relays in cooperative communications: time-switching

relaying protocols and throughput analysis,”IEEE Trans. Commun., vol. 63, no. 5, pp. 1607-1622, May 2015.

[13] I. Krikidis, “Simultaneous information and energy transfer in large-scale networks with/without relaying,”IEEE Trans. Commun.,

vol. 62, no. 3, pp. 900-912, Mar. 2014.

[14] H. Chen, Y. Li, J. L. Rebelatto, B. F. Uchoa-Filho, and B.Vucetic, “Harvest-then-cooperate: wireless-powered cooperative

communications,”IEEE Trans. Signal Process., vol. 63, no. 7, pp. 1700-1711, Apr. 2015.

[15] J. Pan, Y. T. Hou, L. Cai, Y. Shi, and S. X. Shen, “Optimal base-station locations in two-tiered wireless sensor networks,” IEEE Trans.

Mobile Comput., vol. 4, no. 5, pp. 458-473, Sep. 2005.

[16] A. Bogdanov, E. Maneva, and S. Riesenfeld, “Power-aware base station positioning for sensor networks,” inProc. IEEE INFOCOM,

Hong Kong, Mar. 2004.

[17] K. Akkaya, M. Younis, and W. Youssef, “Positioning of base stations in wireless sensor networks,”IEEE Commun. Mag., vol. 45,

no. 4, pp. 96-102, Apr. 2007.

[18] S. R. Gandham, M. Dawande, R. Prakash, and S. Venkatesan, “Energy efficient schemes for wireless sensor networks with multiple

mobile base stations,” inProc. IEEE GLOBECOM, Dec. 2003.

[19] B. Lin, P. H. Ho, L. Xie, X. Shen, and J. Tapolcai, “Optimal Relay Station Placement in Broadband Wireless Access Networks,” IEEE

Trans. Mobile Comput., vol. 9, no. 2, pp. 259-269, Feb. 2010.

[20] A. Goldsmith,Wireless communications, Cambridge University Press, New York, 2005.

[21] Y. T. Hou, Y. Shi, H. D. Sherali, and S. F. Midkiff, “On energy provisioning and relay node placement for wireless sensor networks,”

IEEE Trans. Wireless Commun., vol. 4, no. 5, pp. 2579-2590, Sep. 2005.

[22] X. Liu and P. Mohapatra, “On the deployment of wireless data back-haul networks,”IEEE Trans. Wireless Commun., vol. 6, no. 4,

pp. 1426-1435, Apr 2007.


32

[23] S. Bi and R. Zhang, “Wireless power charging control in multiuser broadband networks,” inProc. IEEE ICC Workshops, London,

U.K., Jun. 2015.

[24] N. Michelusi, L. Badia, R. Carli, L. Corradini, and M. Zorzi, “Energy management policies for harvesting-based wireless sensor

devices with battery degradation,”IEEE Trans. Commun., vol. 61, no. 12, pp. 4934-4947, Dec. 2013.

[25] C. M. Bishop,Pattern recognition and machine learning, Springer, New York, 2006.

[26] J. Hromkovic, Algorithmics for hard problems: introduction to combinatorial optimization, randomization, approximation, and

heuristics, Springer-Verlag, Heidelberg, 2010.

[27] S. Boyd and L. Vandenberghe,Convex optimization, Cambridge University Press, New York, 2004.


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